Is Gravitational Force Considered Properly in an Elasto-Plastic Numerical Analysis of Underground Structures?

"Gravitational force is an external body force acting on underground structures. It should be noted that gravitational force cannot be replaced with the surface traction vector (Cauchy stress) which is an internal force. This fact should be properly taken into account in a numerical analysis of underground structures, particularly for structures such as vertical shafts, tunnels, and caverns constructed at great depths. It is obvious that the gravitational force must be conserved at all times.In conventional numerical analyses, the initial state of stress of the ground medium is firstly calculated under the gravitational field. This means that the gravitational force is replaced with stress (surface traction vector), and after that only the initial stress is considered in the calculation of the stresses and displacements of the structure. It should be noted that the initial stress is Cauchy stress consisting of a surface traction vector, while the gravitational force is a body force acting on the mass of the materials. This computational approach may be acceptable for elastic problems because the principle of superposition is valid for elastic materials. However, this approach becomes questionable for elasto-plastic materials because there is no guarantee that the principle of superposition is valid for them.Furthermore, another problem exists in elasto-plastic analyses: The stress state in a plastic zone must satisfy a yielding criterion such as that of Mohr-Coulomb. In order to make the stress state satisfy the criterion, an excess stress beyond the yielding criterion should be transferred to an elastic zone existing outside of the plastic zone. This seems to always be possible, because an elastic zone always exists, no matter how large of a plastic zone appears. In some cases, however, the excess stress cannot be transferred to the elastic zone because the gravitational force must be conserved. Therefore, it is obvious that the magnitude of stress around underground structures may become large and surpass the yielding criterion at great depths where a large gravitational force exists.In this paper, it is demonstrated that in a conventional elasto-plastic numerical analysis of a vertical shaft at a great depth, all the stress components around the shaft become very small, even though a large gravitational force exists. This strange result may be due to the fact that the gravitational force is not properly taken into consideration in the numerical analysis."